The complement of a graph G is denoted by G. χ(G) denotes the chromatic number and ω(G) the clique number of G. The cycles of odd length at least five are called odd holes and the complements of odd holes are called odd anti-holes. A graph G is called perfect if, for each induced subgraph G of G, χ(G) = ω(G). Classical examples of perfect graphs consist of bipartite graphs, chordal graphs and c...

Let G be a cubic graph with a perfect matching. An edge e of G is a forcing edge if it is contained in a unique perfect matching M , and the perfect matching M is called uniquely forced. In this paper, we show that a 3-connected cubic graphwith a uniquely forced perfect matching is generated from K4 via Y → 1-operations, i.e., replacing a vertex by a triangle, and further a cubic graph with a u...

Let H denote a finite simple hypergraph. The cover ideal of H, denoted by J = J(H), is the monomial ideal whose minimal generators correspond to the minimal vertex covers of H. We give an algebraic method for determining the chromatic number of H, proving that it is equivalent to a monomial ideal membership problem involving powers of J . Furthermore, we study the sets Ass(R/Js) by exploring th...

A graph G is called g-perfect if, for any induced subgraph H of G, the game chromatic number of H equals the clique number of H. A graph G is called g-col-perfect if, for any induced subgraph H of G, the game coloring number of H equals the clique number of H. In this paper we characterize the classes of g-perfect resp. g-col-perfect graphs by a set of forbidden induced subgraphs. Moreover, we ...

We study the weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular code (in particular, with respect to a vertex if the graph is distance-regular). We show how to compute this distribution by the knowledge of the percentage of the colors over the code. For some partial cases of completely regular codes we derive explicit formulas of weig...

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph of G. We prove that a graph is ...

We study the weight distribution of a perfect coloring (equitable partition) of a graph with respect to a completely regular code (in particular, with respect to a vertex if the graph is distance-regular). We show how to compute this distribution by the knowledge of the percentage of the colors over the code. For some partial cases of completely regular codes we derive explicit formulas of weig...

In Defective Coloring we are given a graph $G$ and two integers $\chi_d$, $\Delta^*$ asked if can $\chi_d$-color so that the maximum degree induced by any color class is at most $\Delta^*$. We show this natural generalization of much harder on several basic classes. particular, it NP-hard split graphs, even when one parameters set to smallest possible fixed value does not trivialize problem ($\...